Masses of Ground- & Excited-State Hadrons Craig D. Roberts Physics Division Argonne National Laboratory & School of Physics Peking University Masses of

Masses of Ground- & Excited-State Hadrons

Craig D. Roberts

Physics DivisionArgonne National Laboratory

&

School of PhysicsPeking University

Masses of ground and excited-state hadronsHannes L.L. Roberts, Lei Chang, Ian C. Cloët and Craig D. RobertsarXiv:1101.4244 [nucl-th] , to appear in Few Body Systems

http://inspirebeta.net/record/885274?ln=en



QCD’s Challenges

Dynamical Chiral Symmetry Breaking Very unnatural pattern of bound state masses;

e.g., Lagrangian (pQCD) quark mass is small but . . . no degeneracy between JP=+ and JP=− (parity partners)

Neither of these phenomena is apparent in QCD’s Lagrangian Yet they are the dominant determining characteristics

of real-world QCD.

Both will be important herein QCD

– Complex behaviour arises from apparently simple rules.Craig Roberts, Physics Division: Masses of Ground & Excited State Hadrons

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Quark and Gluon ConfinementNo matter how hard one strikes the proton, one cannot liberate an individual quark or gluon

Hall-B/EBAC: 22 Feb 2011

Understand emergent phenomena

Universal Truths

Spectrum of hadrons (ground, excited and exotic states), and hadron elastic and transition form factors provide unique information about long-range interaction between light-quarks and distribution of hadron's characterising properties amongst its QCD constituents.

Dynamical Chiral Symmetry Breaking (DCSB) is most important mass generating mechanism for visible matter in the Universe.

Higgs mechanism is (almost) irrelevant to light-quarks. Running of quark mass entails that calculations at even modest Q2 require a

Poincaré-covariant approach. Covariance requires existence of quark orbital angular momentum in hadron's rest-frame wave function.

Confinement is expressed through a violent change of the propagators for coloured particles & can almost be read from a plot of a states’ dressed-propagator.

It is intimately connected with DCSB.Craig Roberts, Physics Division: Masses of Ground & Excited State Hadrons

3Hall-B/EBAC: 22 Feb 2011

Universal Conventions

Wikipedia: (http://en.wikipedia.org/wiki/QCD_vacuum)“The QCD vacuum is the vacuum state of quantum chromodynamics (QCD). It is an example of a non-perturbative vacuum state, characterized by many non-vanishing condensates such as the gluon condensate or the quark condensate. These condensates characterize the normal phase or the confined phase of quark matter.”

Craig Roberts, Physics Division: Masses of Ground & Excited State Hadrons


http://en.wikipedia.org/wiki/QCD_vacuum

Universal Misapprehensions

Since 1979, DCSB has commonly been associated literally with a spacetime-independent mass-dimension-three “vacuum condensate.”

Under this assumption, “condensates” couple directly to gravity in general relativity and make an enormous contribution to the cosmological constant

Experimentally, the answer is

Ωcosm. const. = 0.76 This mismatch is a bit of a problem.



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103

8

HG QCDNscondensateQCD

Resolution?

Quantum Healing Central: “KSU physics professor [Peter Tandy] publishes groundbreaking

research on inconsistency in Einstein theory.” Paranormal Psychic Forums:

“Now Stanley Brodsky of the SLAC National Accelerator Laboratory in Menlo Park, California, and colleagues have found a wayto get rid of the discrepancy. “People have just been taking it on faith that this quark condensate is present throughout the vacuum,” says Brodsky.

TAMU Phys. & Astr. Colloquium

Craig Roberts, Physics Division: Much Ado About Nothing

6

Resolution– Whereas it might sometimes be convenient in computational

truncation schemes to imagine otherwise, “condensates” do not exist as spacetime-independent mass-scales that fill all spacetime.

– So-called vacuum condensates can be understood as a property of hadrons themselves, which is expressed, for example, in their Bethe-Salpeter or light-front wavefunctions.

– GMOR cf.

QCD

Paradigm shift:In-Hadron Condensates


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Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201Brodsky and Shrock, arXiv:0905.1151 [hep-th], PNAS 108, 45 (2011)






– No qualitative difference between fπ and ρπ


8TAMU Phys. & Astr. Colloquium

Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201Brodsky and Shrock, arXiv:0905.1151 [hep-th], to appear in PNAS




– No qualitative difference between fπ and ρπ

– And



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0);0( qq

Chiral limit


Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201Brodsky and Shrock, arXiv:0905.1151 [hep-th], to appear in PNAS


“EMPTY space may really be empty. Though quantum theory suggests that a vacuum should be fizzing with particle activity, it turns out that this paradoxical picture of nothingness may not be needed. A calmer view of the vacuum would also help resolve a nagging inconsistency with dark energy, the elusive force thought to be speeding up the expansion of the universe.”


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“Void that is truly empty solves dark energy puzzle”Rachel Courtland, New Scientist 4th Sept. 2010


Cosmological Constant: Putting QCD condensates back into hadrons reduces the mismatch between experiment and theory by a factor of 1046

Possibly by far more, if technicolour-like theories are the correct paradigm for extending the Standard Model

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HG QCDNscondensateQCD

http://www.newscientist.com/article/mg19325911.700-dark-energy-seeking-the-heart-of-darkness.html

QCD and Hadrons

Nonperturbative tools are neededQuark models; Lattice-regularized QCD; Sum Rules;

Generalised Parton Distributions “Theory Support for the Excited Baryon Program at the JLab 12- GeV Upgrade” – arXiv:0907.1901 [nucl-th]

Dyson-Schwinger equations Nonperturbative symmetry-preserving tool

for the study of Continuum-QCD DSEs provide complete and compelling understanding of

the pion as both a bound-state & Nambu-Goldstone mode in QCDCraig Roberts, Physics Division: Masses of Ground & Excited State Hadrons


Gap equation

Pion mass and decay constant, P. Maris, C.D. Roberts and P.C. Tandynucl-th/9707003, Phys. Lett. B20 (1998) 267-273






Dyson-SchwingerEquations

Well suited to Relativistic Quantum Field Theory Simplest level: Generating Tool for Perturbation

Theory . . . Materially Reduces Model-Dependence … Statement about long-range behaviour of quark-quark interaction

NonPerturbative, Continuum approach to QCD Hadrons as Composites of Quarks and Gluons Qualitative and Quantitative Importance of:

Dynamical Chiral Symmetry Breaking– Generation of fermion mass from

nothing Quark & Gluon Confinement

– Coloured objects not detected, Not detectable?Craig Roberts, Physics Division: Masses of Ground & Excited State Hadrons

12

Approach yields Schwinger functions; i.e., propagators and verticesCross-Sections built from Schwinger FunctionsHence, method connects observables with long- range behaviour of the running couplingExperiment ↔ Theory comparison leads to an understanding of long- range behaviour of strong running-coupling


Frontiers of Nuclear Science:Theoretical Advances

In QCD a quark's effective mass depends on its momentum. The function describing this can be calculated and is depicted here. Numerical simulations of lattice QCD (data, at two different bare masses) have confirmed model predictions (solid curves) that the vast bulk of the constituent mass of a light quark comes from a cloud of gluons that are dragged along by the quark as it propagates. In this way, a quark that appears to be absolutely massless at high energies (m =0, red curve) acquires a large constituent mass at low energies.






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DSE prediction of DCSB confirmed

Mass from nothing!





15

Hint of lattice-QCD support for DSE prediction of violation of reflection positivity


12GeVThe Future of JLab

Numerical simulations of lattice QCD (data, at two different bare masses) have confirmed model predictions (solid curves) that the vast bulk of the constituent mass of a light quark comes from a cloud of gluons that are dragged along by the quark as it propagates. In this way, a quark that appears to be absolutely massless at high energies (m =0, red curve) acquires a large constituent mass at low energies.


16

Jlab 12GeV: Scanned by 2<Q2<9 GeV2 elastic & transition form factors.


Gap EquationGeneral Form

Dμν(k) – dressed-gluon propagator Γν(q,p) – dressed-quark-gluon vertex Suppose one has in hand – from anywhere – the exact

form of the dressed-quark-gluon vertex

What is the associated symmetry-preserving Bethe-Salpeter kernel?!



Bethe-Salpeter EquationBound-State DSE

K(q,k;P) – fully amputated, two-particle irreducible, quark-antiquark scattering kernel

Textbook material. Compact. Visually appealing. Correct

Blocked progress for more than 60 years.



Bethe-Salpeter EquationGeneral Form

Equivalent exact bound-state equation but in this form K(q,k;P) → Λ(q,k;P)

which is completely determined by dressed-quark self-energy Enables derivation of a Ward-Takahashi identity for Λ(q,k;P)


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Lei Chang and C.D. Roberts0903.5461 [nucl-th]Phys. Rev. Lett. 103 (2009) 081601





Ward-Takahashi IdentityBethe-Salpeter Kernel

Now, for first time, it’s possible to formulate an Ansatz for Bethe-Salpeter kernel given any form for the dressed-quark-gluon vertex by using this identity

This enables the identification and elucidation of a wide range of novel consequences of DCSB


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Lei Chang and C.D. Roberts0903.5461 [nucl-th]Phys. Rev. Lett. 103 (2009) 081601

iγ5 iγ5





Dressed-quark anomalousmagnetic moments

Schwinger’s result for QED: pQCD: two diagrams

o (a) is QED-likeo (b) is only possible in QCD – involves 3-gluon vertex

Analyse (a) and (b)o (b) vanishes identically: the 3-gluon vertex does not contribute to

a quark’s anomalous chromomag. moment at leading-ordero (a) Produces a finite result: “ – ⅙ αs/2π ”

~ (– ⅙) QED-result But, in QED and QCD, the anomalous chromo- and electro-

magnetic moments vanish identically in the chiral limit!Craig Roberts, Physics Division: Masses of Ground & Excited State Hadrons



Interaction term that describes magnetic-moment coupling to gauge fieldo Straightforward to show that it mixes left ↔ righto Thus, explicitly violates chiral symmetry

Follows that in fermion’s e.m. current γμF1 does cannot mix with σμνqνF2

No Gordon Identityo Hence massless fermions cannot not possess a measurable

chromo- or electro-magnetic moment But what if the chiral symmetry is dynamically

broken, strongly, as it is in QCD?Craig Roberts, Physics Division: Masses of Ground & Excited State Hadrons



Three strongly-dressed and essentially-

nonperturbative contributions to dressed-quark-gluon vertex:


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DCSB

Ball-Chiu term•Vanishes if no DCSB•Appearance driven by STI

Anom. chrom. mag. mom.contribution to vertex•Similar properties to BC term•Strength commensurate with lattice-QCD

Skullerud, Bowman, Kizilersu et al.hep-ph/0303176

Role and importance isNovel discovery•Essential to recover pQCD•Constructive interference with Γ5


L. Chang, Y. –X. Liu and C.D. RobertsarXiv:1009.3458 [nucl-th]Phys. Rev. Lett. 106 (2011) 072001

http://inspirebeta.net/record/868745



http://link.aps.org/doi/10.1103/PhysRevLett.106.072001







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Formulated and solved general Bethe-Salpeter equation Obtained dressed electromagnetic vertex Confined quarks don’t have a mass-shello Can’t unambiguously define

magnetic momentso But can define

magnetic moment distribution

ME κACM κAEM

Full vertex 0.44 -0.22 0.45

Rainbow-ladder 0.35 0 0.048

AEM is opposite in sign but of roughly equal magnitude as ACMo Potentially important for elastic & transition form factors, etc.o Muon g-2 – via Box diagram?


L. Chang, Y. –X. Liu and C.D. RobertsarXiv:1009.3458 [nucl-th]Phys. Rev. Lett. 106 (2011) 072001

Factor of 10 magnification









Dressed Vertex & Meson Spectrum

Splitting known experimentally for more than 35 years Hitherto, no explanation Systematic symmetry-preserving, Poincaré-covariant DSE

truncation scheme of nucl-th/9602012.o Never better than ⅟₄ of splitting∼

Constructing kernel skeleton-diagram-by-diagram, DCSB cannot be faithfully expressed:

Craig Roberts, Physics Division: DSEs for Hadron Physics

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Experiment Rainbow-ladder

One-loop corrected

Ball-Chiu Full vertex

a1 1230

ρ 770

Mass splitting 455

Full impact of M(p2) cannot be realised!

KITP-China: 15 Nov 2010


One-loop corrected


a1 1230 759 885

ρ 770 644 764

Mass splitting 455 115 121



Dressed Vertex & Meson Spectrum

Fully consistent treatment of Ball-Chiu vertexo Retain λ3 – term but ignore Γ4 & Γ5

o Some effects of DCSB built into vertex & Bethe-Salpeter kernel Big impact on σ – π complex But, clearly, not the complete answer.

Fully-consistent treatment of complete vertex Ansatz Promise of 1st reliable prediction of light-quark

hadron spectrumCraig Roberts, Physics Division: DSEs for Hadron Physics

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One-loop corrected


a1 1230 759 885 1066

ρ 770 644 764 924

Mass splitting 455 115 121 142

KITP-China: 15 Nov 2010


One-loop corrected


a1 1230 759 885 1066 1230

ρ 770 644 764 924 745

Mass splitting 455 115 121 142 485

Craig Roberts, Physics Division: Baryon Properties from Continuum-QCD

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I.C. Cloët, C.D. Roberts, et al.arXiv:0812.0416 [nucl-th]

)(

)(

2

2

QG

QG

pM

pEp

Highlights again the critical importance of DCSB in explanation of real-world observables.

Baryons 2010: 11 Dec 2010

DSE result Dec 08

DSE result – including the anomalous magnetic moment distribution

I.C. Cloët, C.D. Roberts, et al.In progress




Radial Excitations

Goldstone modes are the only pseudoscalar mesons to possess a nonzero leptonic decay, fπ ,constant in the chiral limit when chiral symmetry is dynamically broken.

The decay constants of their radial excitations vanish. In quantum mechanics, decay constants are suppressed by a factor of roughly

⅟₃, but only a symmetry can ensure that something vanishes.

Goldstone’s Theorem for non-ground-state pseudoscalars These features and aspects of their impact on the meson

spectrum were illustrated using a manifestly covariant and symmetry-preserving model of the kernels in the gap and Bethe-Salpeter equations.



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Pseudoscalar meson radial excitationsA. Höll, A. Krassnigg & C.D. Roberts, Phys.Rev. C70 (2004) 042203(R), nucl-th/0406030

A Chiral Lagrangian for excited pionsM.K. Volkov & C. WeissPhys. Rev. D56 (1997) 221, hep-ph/9608347



Radial Excitations

These features and aspects of their impact on the meson spectrum were illustrated using a manifestly covariant and symmetry-preserving model of the kernels in the gap and Bethe-Salpeter equations.



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In QFT, too,“wave functions” ofradial excitationspossess a zero

Chiral limitground state = 88MeV

1st excited state: fπ=0



Radial Excitations& Lattice-QCD

When we first heard about [this result] our first reaction was a combination of “that is remarkable” and “unbelievable”

CLEO: τ → π(1300) ντ , fπ(1300) < 8.4 MeVDiehl & Hiller hep-ph/0105194

Lattice-QCD check: 163 x 32,

a~ 0.1fm, 2-flavour, unquenched

fπ(1300) / fπ(140) = 0.078 (93) Full ALPHA formulation required to see

suppression, because PCAC relation is at heart of the conditions imposed for improvement (determining coefficients of irrelevant operators)

The suppression of fπ(1300) is a useful benchmark that can be used to tune and validate lattice QCD techniques that try to determine the properties of excited states mesons. Hall-B/EBAC: 22 Feb 2011


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The Decay constant of the first excited pion from lattice QCD.UKQCD Collaboration (C. McNeile, C. Michael et al.) Phys. Lett. B642 (2006) 244, hep-lat/0607032






DSEs and Baryons


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Dynamical chiral symmetry breaking (DCSB)– has enormous impact on meson properties.Must be included in description and prediction of baryon

properties. DCSB is essentially a quantum field theoretical effect.

In quantum field theory Meson appears as pole in four-point quark-antiquark Green function

→ Bethe-Salpeter Equation Nucleon appears as a pole in a six-point quark Green function

→ Faddeev Equation. Poincaré covariant Faddeev equation sums all possible exchanges

and interactions that can take place between three dressed-quarks Tractable equation is founded on observation that an interaction

which describes colour-singlet mesons also generates nonpointlike quark-quark (diquark) correlations in the colour-antitriplet channel

R.T. Cahill et al.,Austral. J. Phys. 42 (1989) 129-145

Hall-B/EBAC: 22 Feb 2011rqq ≈ rπ


Faddeev Equation


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Linear, Homogeneous Matrix equationYields wave function (Poincaré Covariant Faddeev Amplitude)

that describes quark-diquark relative motion within the nucleon Scalar and Axial-Vector Diquarks . . .

Both have “correct” parity and “right” masses In Nucleon’s Rest Frame Amplitude has

s−, p− & d−wave correlations

R.T. Cahill et al.,Austral. J. Phys. 42 (1989) 129-145

diquark

quark

quark exchangeensures Pauli statistics



Unification of Meson & Baryon Spectra

Correlate the masses of meson and baryon ground- and excited-states within a single, symmetry-preserving framework Symmetry-preserving means:

Poincaré-covariant & satisfy relevant Ward-Takahashi identities Constituent-quark model has hitherto been the most widely applied

spectroscopic tool; and whilst its weaknesses are emphasized by critics and acknowledged by proponents, it is of continuing value because there is nothing better that is yet providing a bigger picture.

Nevertheless, no connection with quantum field theory & certainly not with QCD not symmetry-preserving & therefore cannot veraciously connect

meson and baryon properties



33


Dyson-Schwinger Equations have been applied extensively to the spectrum and interactions of mesons with masses less than 1 GeV

On this domain the rainbow-ladder approximation, which is the leading-order in a systematic & symmetry-preserving truncation scheme – nucl-th/9602012, is an accurate, well-understood tool: e.g., Prediction of elastic pion and kaon form factors: nucl-th/0005015 Anomalous neutral pion processes – γπγ & BaBar anomaly: 1009.0067 [nucl-th] Pion and kaon valence-quark distribution functions: 1102.2448 [nucl-th] Unification of these and other observables – ππ scattering: hep-ph/0112015

It can readily be extended to explain properties of the light neutral pseudoscalar mesons: 0708.1118 [nucl-th]



34








http://arxiv.org/abs/1102.2448









Some people have produced a spectrum of mesons with masses above 1GeV – but results have always been poor For the bulk of such studies since 2004, this was a case of

“Doing what can be done, not what needs to be done.” Now understood why rainbow-ladder is not good for states

with material angular momentum know which channels are affected – scalar and axial-vector; and the changes to expect in these channels

Task – Improve rainbow-ladder for mesons & build this knowledge into Faddeev equation for baryons, because formulation of Faddeev equation rests upon knowledge of quark-quark scattering matrix



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Faddeev Equation



36

quark-quark scattering matrix - pole-approximation used to arrive at Faddeev-equation

Rainbow-ladder gap and Bethe-Salpeter equations

In this truncation, colour-antitriplet quark-quark correlations (diquarks) are described by a very similar homogeneous Bethe-Salpeter equation

Only difference is factor of ½ Hence, an interaction that describes mesons also generates

diquark correlations in the colour-antitriplet channel

Diquarks



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Calculation of diquark masses in QCDR.T. Cahill, C.D. Roberts and J. PraschifkaPhys.Rev. D36 (1987) 2804





Vector-vector contact interaction

mG is a gluon mass-scale – dynamically generated in QCD

Gap equation:

DCSB: M ≠ 0 is possible so long as mG<mGcritical

Studies of π & ρ static properties and π form factor establish that contact-interaction results are not realistically distinguishable from those of renormalisation-group-improved one-gluon exchange for Q2<M2

Interaction Kernel



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Studies of π & ρ static properties and π form factor establish that contact-interaction results are not realistically distinguishable from those of renormalisation-group-improved one-gluon exchange for Q2<M2

Interaction Kernel



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contact interaction

QCD 1-loop RGI gluon

M 0.37 0.34

κπ 0.24 0.24

mπ 0.14 0.14

mρ 0.93 0.74

fπ 0.10 0.093

fρ 0.13 0.15

Difference owes primarily to mismatch in mρ

M2

cf. expt. rms rel.err.=13%



contact interaction

M 0.37

κπ 0.24

mπ 0.14

mρ 0.93

fπ 0.10

fρ 0.13

Contact interaction is not renormalisable Must therefore introduce

regularisation scheme Use confining proper-time definition

Λir = 0.24GeV, τir = 1/Λir = 0.8fm

a confinement radius, which is not varied Two parameters:

mG=0.13GeV, Λuv=0.91GeV

fitted to produce tabulated results

Interaction Kernel- Regularisation Scheme



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D. Ebert, T. Feldmann and H. Reinhardt, Phys. Lett. B 388 (1996) 154.No pole in propagator – DSE realisation of confinement



Regularisation & Symmetries

In studies of hadron spectrum it’s critical that an approach satisfy the vector and axial-vector Ward-Takahashi identities. Without this it is impossible to preserve the pattern of chiral symmetry

breaking in QCD & hence a veracious understanding of hadron mass splittings is not achievable.

Contact interaction should & can be regularised appropriately Example: dressed-quark-photon vertex

Contact interaction plus rainbow-ladder entails general form

Vector Ward-Takahashi identity

With symmetry-preserving regularisation of contact interaction, identity requires



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)2/()2/();( 11 QkSQkSQkiQ

)()();( 22 QPQPQk LL

TT

PL(Q2)=1 & PT(Q2=0)=1Interactions cannot generate an on-shell mass for the photon.

Regularisation & Symmetries

Solved Bethe-Salpeter equation for dressed-quark photon vertex, regularised using symmetry-preserving scheme



Ward-Takahashi identity

ρ-meson polegenerated dynamically- Foundation for VMD

“Asymptotic freedom”Dressed-vertex → bare at large spacelike Q2

RGI one-gluon exchangeMaris & Tandy prediction of Fπ(Q2)

Bethe-Salpeter Equations

Ladder BSE for ρ-meson

ω(M2,α,P2)= M2 + α(1- α)P2

Contact interaction, properly regularised, provides a practical simplicity and physical transparency

Ladder BSE for a1-meson

All BSEs are one- or –two dimensional eigenvalue problems, eigenvalue is P2= - (mass-bound-state)2



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)()(),,0(),0()( 112222

1 iuiu

iruv

iuCCrMrMC

Meson Spectrum-Ground-states Ground-state masses

Computed very often, always with same result

Namely, with rainbow-ladder truncation

ma1 – mρ = 0.15 GeV ≈ ⅟₃ x 0.45experiment



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One-loop corrected

Full vertex

a1 1230 759 885 1230

ρ 770 644 764 745

Mass splitting 455 115 121 485

But, we know how to fix that viz., DCSB – a beyond rainbow ladderincreases scalar and axial-vector massesleaves π & ρ unchanged

Meson Spectrum-Ground-states Ground-state masses

Correct for omission of DCSB-induced spin-orbit repulsion

Leave π- & ρ-meson BSEs unchanged but introduce repulsion parameter in scalar and axial-vector channels; viz.,

gSO=0.24 fitted to produce ma1 – mρ = 0.45experiment



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2SOg

mσqq ≈ 1.2 GeV is location of

quark core of σ-resonance:• Pelaez & Rios (2006)• Ruiz de Elvira, Pelaez, Pennington & Wilson (2010)First novel post-diction

Meson Spectrum- Radial Excitations

As illustrated previously, radial excitations possess a single zero in the relative-momentum dependence of the leading Tchebychev-moment in their Bethe-Salpeter amplitude

The existence of radial excitations is therefore very obvious evidence against the possibility that the interaction between quarks is momentum-independent: A bound-state amplitude that is independent of the relative momentum cannot

exhibit a single zero One may express this differently; viz.,

If the location of the zero is at k02, then a momentum-independent interaction

can only produce reliable results for phenomena that probe momentum scales k2 < k0

2.

In QCD, k0 ≈ M.



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Meson Spectrum- Radial Excitations

Nevertheless, there exists an established expedient ; viz., Insert a zero by hand into the Bethe-Salpeter kernels

Plainly, the presence of this zero has the effect of reducing the coupling in the BSE & hence it increases the bound-state's mass.

Although this may not be as transparent with a more sophisticated interaction, a qualitatively equivalent mechanism is always responsible for the elevated values of the masses of radial excitations.

Location of zero fixed at “natural” location – not a parameter



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* *

(1-k2/M2)

A Chiral Lagrangian for excited pionsM.K. Volkov & C. WeissPhys. Rev. D56 (1997) 221, hep-ph/9608347

Meson SpectrumGround- & Excited-States

Complete the table …

Error estimate for radial excitations: Shift location of zero by ±20%

rms-relative-error/degree-of-freedom = 13%

No parameters Realistic DSE estimates: m0+=0.7-0.8, m1+=0.9-1.0 Lattice-QCD estimate: m0+=0.78 ±0.15, m1+-m0+=0.14



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plus predicted diquark spectrum

NO results for other qq quantum numbers,critical for excited statesof N and Δ

Spectrum of Baryons

Static “approximation” Implements analogue of contact interaction in Faddeev-equation

In combination with contact-interaction diquark-correlations, generates Faddeev equation kernels which themselves are momentum-independent

The merit of this truncation is the dramatic simplifications which it produces

Used widely in hadron physics phenomenology; e.g., Bentz, Cloët, Thomas et al.



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M

g

MpiB21

Variant of:A. Buck, R. Alkofer & H. Reinhardt, Phys. Lett. B286 (1992) 29.

Spectrum of Baryons

Static “approximation” Implements analogue of contact interaction in Faddeev-equation

From the referee’s report:In these calculations one could argue that the [static truncation] is the weakest [approximation]. From what I understand, it is not of relevance here since the aim is to understand the dynamics of the interactions between the [different] types of diquark correlations with the spectator quark and their different contributions to the baryon's masses … this study illustrates rather well what can be expected from more sophisticated models, whether within a Dyson-Schwinger or another approach. … I can recommend the publication of this paper without further changes.



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M

g

MpiB21

Spectrum of Baryons



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M

g

MpiB21

Faddeev equation for Δ-resonance

One-dimensional eigenvalue problem, to which only axial-vector diquark contributes

Nucleon has scalar & axial-vector diquarks. It is a three-dimensional eigenvalue problem

Spectrum of Baryons“pion cloud”



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M

g

MpiB21

Pseudoscalar-meson loop-corrections to our truncated DSE kernels may have a material impact on mN and mΔ separately but the

contribution to each is approximately the same so that the mass-difference is largely unaffected by such

corrections: (mΔ- mN)π-loops= 40MeV

1. EBAC: “undressed Δ” has mΔ = 1.39GeV;

2. (mΔ- mN)qqq-core= 250MeV

achieved with gN=1.18 & gΔ=1.56

All three spectrum parameters now fixed (gSO=0.24)

Baryons & diquarks



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Provided numerous insights into baryon structure; e.g., There is a causal connection between mΔ - mN & m1+- m0+

mΔ - mN

mN

mΔ

Physical splitting grows rapidly with increasing diquark mass difference

Baryons & diquarks



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Provided numerous insights into baryon structure; e.g., mN ≈ 3 M & mΔ ≈ M+m1+

Baryon Spectrum



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Our predictions for baryon dressed-quark-core masses match the bare-masses determined by Jülich with a rms-relative-error of 10%. Notably, however, we find a quark-core to the Roper resonance,

whereas within the Jülich coupled-channels model this structure in the P11 partial wave is unconnected with a bare three-quark state.

Baryon Spectrum



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In connection with EBAC's analysis, our predictions for the bare-masses agree within a rms-relative-error of 14%. Notably, EBAC does find a dressed-quark-core for the Roper

resonance, at a mass which agrees with our prediction.

Hadron Spectrum



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Legend: Particle Data Group H.L.L. Roberts et al. EBAC Jülich

o Symmetry-preserving unification of the computation of meson & baryon masseso rms-rel.err./deg-of-freedom = 13%o PDG values (almost) uniformly overestimated in both cases - room for the pseudoscalar meson cloud?!

Next steps…

DSE treatment of static and electromagnetic properties of pseudoscalar and vector mesons, and scalar and axial-vector diquark correlations based upon a vector-vector contact-interaction.

Basic motivation: need to document a comparison between the electromagnetic form factors of mesons and those diquarks which play a material role in nucleon structure. Important step toward a unified description of meson and baryon form factors based on a single interaction.

Notable results: o Large degree of similarity between related meson and diquark form factors. o Zero in the ρ-meson electric form factor at zQ

ρ ≈ √6 mρ .

Notably, r ρ zQρ ≈ rD zQ

D, where r ρ, rD are, respectively, the electric radii of the ρ-meson and deuteron.

Ready now for nucleon elastic & nucleon→Roper transition form factors



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Epilogue

Dynamical chiral symmetry breaking (DCSB) – mass from nothing for 98% of visible matter – is a realityo Expressed in M(p2), with observable signals in experiment

Poincaré covarianceCrucial in description of contemporary data

Fully-self-consistent treatment of an interaction Essential if experimental data is truly to be understood.

Dyson-Schwinger equations: o single framework, with IR model-input turned to advantage,

“almost unique in providing unambiguous path from a defined interaction → Confinement & DCSB → Masses → radii → form factors → distribution functions → etc.” McLerran & Pisarski

arXiv:0706.2191 [hep-ph]

Confinement is almost Certainly the origin of DCSB





Documents

Masses of Ground- & Excited-State Hadrons Craig D. Roberts Physics Division Argonne National Laboratory & School of Physics Peking University Masses of